My book stated the following theorem: If the functions P(x) and Q(x) are continuous on the open interval I containing the point x(adsbygoogle = window.adsbygoogle || []).push({}); _{0}, then the initial value problem dy/dx + P(x)y = Q(x), y(x_{0})=y_{0}has a unique solution y(x) on I, given by the formula y=1/I(x)[itex]\int[/itex]I(x)Q(x)dx where I(x) is the integrating factor.

Now the book showed how the Integrating Factor Method was developed, but it doesn't prove this theorem, particularly why a unique solution exists and why there are no other solutions of a different form (singular solutions).

Also it states, "The appropriate value of the constant C can be selected "automatically" by writing, I(x)=exp([itex]\int[/itex][itex]^{x_{0}}_{x}[/itex]P(t)dt) and y(x)=1/I(x)[y_{0}+[itex]\int[/itex][itex]^{x_{0}}_{x}[/itex]I(t)Q(t)dt]"

I don't understand how they got this form. If you have a definite integral there shouldn't be constant like y_{0}either...

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# Integrating factor for first order linear equations uniqueness theorem

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